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\title{Shapes of crystals}
\author{P. Pansu, Université Paris-Sud 11 and Ecole Normale Supérieure}
\date{June 30th, 2010}

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\begin{document}

\begin{frame}
\titlepage
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\section{Jeux d'enfants}

\begin{frame}

On a game board, form a shape using $A$ identical pieces. The {\em perimeter} is the number of outer edges, i.e. edges which are not shared by two pieces. The number $A$ is the {\em area} of the shape.
\begin{center}
\includegraphics[width=1.15in]{jeucarres.pdf}\includegraphics[width=1.5in]{jeutriangles.pdf}\includegraphics[width=1.2in]{jeuhexagones.pdf}
\end{center}

\end{frame}

\begin{frame}
On a game board, form a shape using $A$ identical pieces. The {\em perimeter} is the number of outer edges, i.e. edges which are not shared by two pieces. The number $A$ is the {\em area} of the shape.
\begin{center}
\includegraphics[width=1.15in]{jeucarres2.pdf}\includegraphics[width=1.5in]{jeutriangles2.pdf}\includegraphics[width=1.2in]{jeuhexagones2.pdf}
\end{center}
\pause

Aim of the game : find the shape which, for a given area, minimizes perimeter.

\bigskip

These are three {\em discrete} isoperimetric problems. 

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\section{Crystals}

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Here are a few solutions of isoperimetric games.

\begin{center}
\includegraphics[width=1.15in]{jeucarres3.pdf}\includegraphics[width=1.5in]{jeutriangles3.pdf}\includegraphics[width=1.2in]{jeuhexagones3.pdf}
\end{center}\pause

\vskip-.5cm

\hbox{\vbox{\hsize=7cm When area gets larger, solutions have the same rough shapes, approximately square or hexagonal. 
}
\vtop{\includegraphics[width=1in,angle=90]{jeuhexagones4.pdf}}}

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\begin{frame}

\begin{thm}
{\em (1999)}. Three dimensional variants of these games yield similar results:  large volume solutions approximate polyhedra. 
\end{thm}\pause

This reminds of crystals found in nature.
\begin{center}
\includegraphics[width=.8in]{cube_fer.jpg} \hskip.5cm \includegraphics[width=.7in]{octaedre.jpg}\hskip.5cm \includegraphics[width=.8in]{brigitte.jpg}\hskip.5cm \includegraphics[width=.8in]{664.jpg}
\end{center}\pause

\hbox{\vbox{\hsize=6cm R.J. Haüy (1784) proposed that matter should be formed of tiny polyhedra glued together. This was naive, but the idea of microscopic periodicity was a good one, and finally was confirmed experimentally in 1912.}
\vtop{\includegraphics[width=1.3in]{Hauy.pdf}}}

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\section{Quasicrystals}

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\begin{prop}
{\em Crystallographic restriction}. In $2$ or $3$ dimensions, if a periodic array of points is stabilized by a rotation or screw rotation, the rotation angle must be a multiple of $\pi/3$ or $\pi/2$.
\end{prop}\pause

\textbf{Proof} {\em (Hauÿ, 1822)}. In a basis of translation vectors stabilizing the array, the matrix of the rotation has integral entries. Its trace is an integer. Or, in an arbitrary basis, the trace of a rotation with angle $\theta$ equals $1+2\cos(\theta)$. Thus $\cos(\theta)=1,\frac{1}{2},0,-\frac{1}{2},-1$, $\theta=0,\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\pi$.\pause


\bigskip

During 200 years, crystallographers have taken this rule for granted. It has been confirmed by {\em diffraction patterns} (obtained using X-rays), which reflect the crystal's symmetry.

\begin{center}
\includegraphics[width=1in]{diffraction_carre.jpg}\hskip1cm\includegraphics[width=1.4in]{diffraction_decr5a-a.jpg}
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\begin{frame}

How can one explain the $5$-fold symmetry visible in the diffraction pattern below ? And the dodecahedral shape of a monocrystal of aluminium-copper-iron alloy ?
\begin{center}
\includegraphics[width=1in]{diffraction_10_semsci_strasbg.pdf}\hskip1.5cm\includegraphics[width=0.875in]{AlCuFe_dodeca.pdf}
\end{center}\pause

One must admit that crystalline state is not characterized by periodicity, but by other properties. In 1992, the International Crystallographic Union chose the following definition.
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{\em A crystal is a solid whose diffraction pattern is made of distinct spots.}
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Quasiperiodicity can produce crystals. A phenomenon in $3$-space is {\em quasiperiodic} if it is the trace, on a $3$-dimensional vector subspace, of a higher dimensional periodic phenomenon.

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\begin{frame}
\hbox{\vbox{\hsize=4.5cm To obtain a non periodic tiling of a line, one can consider an oblique slab in the plane tiled by squares, and project on a line the vertices contained in the slab.
}
\vtop{\hskip1cm \includegraphics[width=1.3in]{droite.pdf}}}\pause

\bigskip

\hbox{\vbox{\hsize=4.5cm This non periodic tiling has been obtained as follows. Consider an oblique slab in $4$-dimensional space tiled by hypercubes, and project on a plane the vertices contained in the slab.
 }
\vtop{\hskip1.3cm \includegraphics[width=1in]{Penrose.jpg}}}
\pause 

\bigskip

The same process applied in dimension $6$ generates non periodic tilings of $3$-dimensional space, whose diffraction patterns are made of spots, and which admit screw rotations with angle $2\pi/5$. These are good candidates for models of {\em quasicrystals}.

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